Referring to the distribution of the sequence of factorials modulo a prime \(p\) , an earlier simpler
problem was proposed by P. Erdős [1]}. He asks whether there
exists \(p>5\)  for which the numbers \(2!, 3!,\dots , (p-1)!\)  are all distinct modulo \(p\) .
If there were such a prime, then T. Trudgian [2]}
verified that it must be greater than \(10^9\) . The problem of Erdős is still unsolved,
although B. Rokowska, A. Schinzel [3]}
and Trudgian [2]} showed that for large classes of primes the sequence of
factorials modulo \(p\)  can not be so close to a permutation, while Klurman and Munsch [5]}
obtained non-trivial bounds for the average deviation.
